What is convergent matrix?

A convergent matrix is a square matrix that converges to a limit matrix as the number of matrix multiplications approaches infinity. In other words, a matrix is said to be convergent if its power series tends towards a limit matrix. This means that the values of each element in the matrix will eventually settle down to a constant value after a certain number of iterations.

A convergent matrix is also known as a stable matrix or a regular matrix. It is used in applications like Markov chains and iterative methods in numerical analysis. Convergent matrices are important in the study of dynamical systems and chaos theory.

To determine whether a matrix is convergent, we can use the spectral radius test. If the spectral radius of the matrix is less than 1, then the matrix is convergent. Alternatively, if the matrix is diagonalizable, we can calculate its eigenvalues. If all of the eigenvalues have absolute values less than 1, then the matrix is convergent.

Examples of convergent matrices include the identity matrix and the following matrix:

M = [0.8 0.2; 0.4 0.6]

The spectral radius of M is 0.7, which is less than 1. Therefore, M is a convergent matrix.